On the Incompleteness of (k, n)-arcs in Desarguesian Planes of Order q Where n Divides q

We investigate the completeness of an ( nq – q + n – , n)-arc in the Desarguesian plane of order q where n divides q. It is shown that such arcs are incomplete for 0< n/2 if q/n3. For q = 2n they are incomplete for 0 < < 0.381n and for q = 3n they are incomplete for 0 < < 0.476n. For q odd it is known that such arcs do not exist for = 0 and, hence, we improve the upper bound on the maximum size of such a ( k, n)-arc.     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersM _k such that, for anyf(z)c[0,1] and > 0 there exists the polynomial that f–P< and _k <M _k ,k=1, ...,n.Translated from Matematicheskii Zametki, Vol. 22, No. 2, pp. 269–276, August, 1977.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Lower semicontinuity of attractors of gradient systems and applications

Summary For 0₀, let T(t), t0, be a family of semigroups on a Banach space X with local attractors A. Under the assumptions that T₀(t) is a gradient system with hyperbolic equilibria and T(t) converges to T₀(t) in an appropriate sense, it is shown that the attractors {A, 0₀} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between A and A₀, in some examples.Research supported by U.S. Army Research Office DAAL-03-86-K-0074 and the National Science Foundation DMS-8507056.     

Asymptotics of the first correction in the perturbation of theN-soliton solution to the KdV equation

We consider a triple Fourier-type integral that represents a solution to the KdV equation linearized on anN-soliton potential. Assuming that the parameters of the potential depend on the slow timet, we construct an asymptotics of this integral as 0 uniform with respect tox, t up to large timet ^–1.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 204–217, August, 1995.The work was financially supported in part by the Russian Foundation for Basic Research under grant No. 94-01-00193a.     

Über die Anzahl der Lösungen der diskreten Theodorsen-Gleichung

Summary Discretization of the Theodorsen integral equation (T) yields the discrete Theodorsen-equation (T _d ), a system of 2N nonlinear equations. A so-called -condition may be fulfilled. It is known that (T) has exactly one continuous solution. This solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domainG. It is also known that (T _d ) has one and only one solution if <1 and at least one solution if 1. We show here that for every 1 and N\ {1} there is a domainG satisfying an -condition such that (T _d ) has an infinite number of solutions. Moreover, givenK>0 and any domainG that fulfills an -condition, we will construct a domainG ₁ in the neighbourhood ofG that fulfills a max (1, +K)-condition such that (T _d ) forG ₁ has an infinite number of solutions. The underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of (T _d ), even in the case <1.

     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

Inverse power law potentials in rectangular configurations

Summary Calculations based on a (distance)^– intermolecular potential (>3) enable study of the effects on adsorption of the geometry of the solid. This paper gives the closed form solution for the adsorptive potential about a homogeneous solid rectangular corner; and, through systematic superposition, closed form solutions for the following configurations also: the rectangular corner of a cavity; laminae and rectangular cracks occupying a quarter plane; semi-infinite rectangular prisms and prismatic cavities; rectangular parallelepipeds and brick-shaped cavities. These various results are developed in detail for the cases =6 and =4. The paradox that potentials for >3 seem to be obtainable more readily than Newtonian potentials (=1) is explained by the existence only for >3 of simple fundamental solutions for infinite homogeneous solid configurations.

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Zusammenfassung Berechnungen, denen ein intermolekulares Potential der Form (Abstand)^– (>3) zugrunde gelegt ist, ermöglichen eine Untersuchung von Effekten der Adsorption auf die Geometrie des Festkörpers. Die vorliegende Arbeit gibt die Lösung in geschlossener Form für das Adsorptionspotential um eine feste, homogene, rechtwinklige Ecke an. Ausserdem werden durch systematische Superposition Lösungen in geschlossener Form für die folgenden Konfigurationen angegeben: die rechtwinklige Innenecke einer Mulde; viertelunendliche, ebene Platten und rechteckige Spalten; halbunendliche, reckteckige Prismen und prismatische Mulden; Quader und quaderförmige Höhlen. Diese Ergebnisse sind ausführlich dargestellt für die Fälle =4. Das Paradoxon. dass Potentiale mit >3 scheinbar leichter zugänglich sind als das Gravitationspotential (=1), wird dadurch erklärt, dass nur für >3 einfache Grundlösungen für unendliche, homogene Festköperkonfigurationen existieren.

     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = × Z of the formU (, z)=(T,, z+f ()) ( , z Z), where is a complete separable metric space, T is a t.t. homeomorphism of onto itself, Z is a separable banach space, andf is a continuous map: z. For the special case W = S¹×R, T=+ ( is incommensurable with 2) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S¹R for which the corresponding homeomorphism (1) is t.t. for all incommensurable with 2.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.The author thanks D. V. Anosov for advice and interest in the work.     

Large deviation principle for the diffusion-transmutation processes and dirichlet problem for PDE systems with small parameter

Summary The diffusion-transmutation processes are considered as the diffusivities are of order ,0 and the transmutation intensities are of order ^–1. We prove a large deviation principle for the position joint with the type occupation times as 0 and study the exit problem for this process. We consider the Levinson case where a trajectory of the average drift field exits from a domain in finite time in a regular way and the large deviation case where the average drift field on the boundary points inward at the domain. The exit place and the type distribution at the exit time are determined as 0; this gives the limit of the Dirichlet problems for the corresponding PDE systems with a parameter 0.Supported in part by ARO Grant DAAL03-92-G0219Supported in part by NSF DMS9207928     

Remarks on the perturbation of analytic matrix functions III

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues () of the perturbed matrix pencilT(, )=A()+B(, ) in the neighbourhood of an unperturbed eigenvalue () ofA(). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue ₀ ofA().     

Über Abstand 1 erhaltende Abbildungen in Minkowski-Ebenen

By using a computer the following theorem is proved: Consider K=GF(q), q {32,64,81,128}, :K² K² bijective such that P,Q K². Then is a semi-isometry. The assumption bijective can be dropped if q {32,128}.     

Semilattices of left reversible semigroups

Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

An algebraic approach to the vector ε-algorithm

The vector -algorithm is obtained from the scalar -algorithm by taking the pseudo-inverse of a vector instead of the inverse of a scalar. Thus the vector -algorithm is known only through its rules contrarily to the scalar -algorithm and some other extrapolation algorithms.The aim of this paper is to provide an algebraic approach to the vector -algorithm.     

Optimization of adaptive algorithms for the renewal of monotone functions from the classHω

A problem of renewal of monotone functionsf(t) H [a, b] with fixed values at the ends of an interval is studied by using adaptive algorithms for calculating the values off(t) at certain points. Asymptotically exact estimates unimprovable on the entire set of adaptive algorithms are obtained for the least possible numberN() of steps providing the uniform-error. For moduli of continuity of type , 0<<1, the valueN() has a higher order as0 than in the nonadaptive case for the same amount of information.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1627–1634, December, 1993.The work was supported by the Foundation for Fundamental Studies of Ukrainian State Committee for Science and Technology.     

Factorization intok-dimensional linear bijections

LetV be a vector space,k withkdimV andS _k{GL(V)|dimV(–1)=k}. ThenS _k generates GL _f (V){GL(V)|V(-1) is finite-dimensional} (with the exception that dimV=2=k and the field is GF2). We study the length problem in GL _f (V) withS _k as set of generators.     

Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.